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Verification of Qualitative ℤ Constraints

  • Stéphane Demri
  • Régis Gascon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3653)

Abstract

We introduce an LTL-like logic with atomic formulae built over a constraint language interpreting variables in ℤ. The constraint language includes periodicity constraints, comparison constraints of the form x = y and x < y, it is closed under Boolean operations and it admits a restricted form of existential quantification. This is the largest set of qualitative constraints over ℤ known so far, shown to admit a decidable LTL extension. Such constraints are those used for instance in calendar formalisms or in abstractions of counter automata by using congruences modulo some power of two. Indeed, various programming languages perform arithmetic operators modulo some integer. We show that the satisfiability and model-checking problems (with respect to an appropriate class of constraint automata) for this logic are decidable in polynomial space improving significantly known results about its strict fragments. As a by-product, LTL model-checking over integral relational automata is proved complete for polynomial space which contrasts with the known undecidability of its CTL counterpart.

Keywords

Model Check Temporal Logic Atomic Formula Symbolic Model Polynomial Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [AD94]
    Alur, R., Dill, D.: A theory of timed automata. TCS 126, 183–235 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  2. [AH94]
    Alur, R., Henzinger, T.: A really temporal logic. JACM 41(1), 181–204 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [BC02]
    Balbiani, P., Condotta, J.F.: Computational complexity of propositional linear temporal logics based on qualitative spatial or temporal reasoning. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309, pp. 162–173. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. [BEH95]
    Bouajjani, A., Echahed, R., Habermehl, P.: On the verification problem of nonregular properties for nonregular processes. In: LICS 1995, pp. 123–133 (1995)Google Scholar
  5. [BEM97]
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: application to model-checking. In: CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)Google Scholar
  6. [BFLP03]
    Bardin, S., Finkel, A., Leroux, J., Petrucci, L.: FAST: Fast Acceleration of Symbolic Transition systems. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 118–121. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. [Boi98]
    Boigelot, B.: Symbolic methods for exploring infinite state spaces. PhD thesis, Université de Liège (1998) Google Scholar
  8. [Cau03]
    Caucal, D.: On infinite transition graphs having a decidable monadic theory. TCS 290, 79–115 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. [CC00]
    Comon, H., Cortier, V.: Flatness is not a weakness. In: Clote, P.G., Schwichtenberg, H. (eds.) CSL 2000. LNCS, vol. 1862, pp. 262–276. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  10. [ˇCer94]
    Čerāns, K.: Deciding properties of integral relational automata. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 35–46. Springer, Heidelberg (1994)Google Scholar
  11. [CGL94]
    Clarke, E., Grumberg, O., Long, D.: Model checking and abstraction. ACM Transactions on Programming Languages and Systems 16(5), 1512–1542 (1994)CrossRefGoogle Scholar
  12. [CJ98]
    Comon, H., Jurski, Y.: Multiple counters automata, safety analysis and Presburger arithmetic. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 268–279. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. [DD03]
    Demri, S., D’Souza, D.: An automata-theoretic approach to constraint LTL. Technical Report LSV-03-11, 40 pages, LSV (August 2003); An extended abstract appeared in Agrawal, M., Seth, A.K. (eds.): FSTTCS 2002. LNCS, vol. 2556. Springer, Heidelberg (2002)Google Scholar
  14. [Dem04]
    Demri, S.: LTL over integer periodicity constraints. Technical Report LSV-04-6, LSV, 35 pages (February 2004); An extended abstract appeared in Walukiewicz, I. (ed.): FOSSACS 2004. LNCS, vol. 2987. Springer, Heidelberg (2004)Google Scholar
  15. [DG05]
    Demri, S., Gascon, R.: Verification of qualitative Z-constraints. Technical Report LSV-05-07, LSV (June 2005) Google Scholar
  16. [EFM99]
    Esparza, J., Finkel, A., Mayr, R.: On the verification of broadcast protocols. In: LICS 1999, pp. 352–359 (1999)Google Scholar
  17. [FL02]
    Finkel, A., Leroux, J.: How to compose Presburger accelerations: Applications to broadcast protocols. In: FST&TCS 2002. LNCS, vol. 2256, pp. 145–156. Springer, Heidelberg (2002)Google Scholar
  18. [GK03]
    Gastin, P., Kuske, D.: Satisfiability and model checking for MSOdefinable temporal logics are in PSPACE. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 222–236. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  19. [GKK+03]
    Gabelaia, D., Kontchakov, R., Kurucz, A., Wolter, F., Zakharyaschev, M.: On the computational complexity of spatio-temporal logics. In: FLAIRS 2003, pp. 460–464 (2003)Google Scholar
  20. [Iba78]
    Ibarra, O.: Reversal-bounded multicounter machines and their decision problems. JACM 25(1), 116–133 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  21. [JKMS04]
    Jančar, P., Kučera, A., Moller, F., Sawa, Z.: DP lower bounds for equivalence-checking and model-checking of one-counter automata. I & C 188, 1–19 (2004)zbMATHGoogle Scholar
  22. [LM01]
    Dal Lago, U., Montanari, A.: Calendars, time granularities, and automata. In: Jensen, C.S., Schneider, M., Seeger, B., Tsotras, V.J. (eds.) SSTD 2001. LNCS, vol. 2121, pp. 279–298. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  23. [LS01]
    Logothetis, G., Schneider, K.: Abstraction from counters: an application on real-time systems. In: TIME 2001, pp. 214–223. IEEE, Los Alamitos (2001)Google Scholar
  24. [Lut04]
    Lutz, C.: NEXPTIME-complete description logics with concrete domains. ACM Transactions on Computational Logic 5(4), 669–705 (2004)CrossRefMathSciNetGoogle Scholar
  25. [MOS05]
    Müller-Olm, M., Seidl, H.: Analysis of modular arithmetic. In: ESOP 2005, LNCS, Springer, Heidelberg (2005)CrossRefGoogle Scholar
  26. [Saf89]
    Safra, S.: Complexity of Automata on Infinite Objects. PhD thesis, The Weizmann Institute of Science (1989) Google Scholar
  27. [SC85]
    Sistla, A., Clarke, E.: The complexity of propositional linear temporal logic. JACM 32(3), 733–749 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  28. [TC98]
    Toman, D., Chomicki, J.: Datalog with integer periodicity constraints. Journal of Logic Programming 35(3), 263–290 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  29. [VW94]
    Vardi, M., Wolper, P.: Reasoning about infinite computations. I & C 115, 1–37 (1994)zbMATHMathSciNetGoogle Scholar
  30. [Wol83]
    Wolper, P.: Temporal logic can be more expressive. I & C 56, 72–99 (1983)zbMATHMathSciNetGoogle Scholar
  31. [WZ00]
    Wolter, F., Zakharyaschev, M.: Spatio-temporal representation and reasoning based on RCC-8. In: KR 2000, pp. 3–14 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Stéphane Demri
    • 1
  • Régis Gascon
    • 1
  1. 1.LSV/CNRS UMR 8643 & INRIA Futurs projet SECSI & ENS CachanCachan CedexFrance

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